Question

How do you solve `sqrt(5x^2-9)=2x` and check your solution?

Answer 1

`x=+3`
(see below for solution and check)

Explanation:

Given
`color(white)("XXX")sqrt(5x^2-9)=2x`

Squaring both sides (this may introduce extraneous solutions; we will need to check later)
`color(white)("XXX")5x^2-9=4x^2`

Subtract `4x^2` from both sides
`color(white)("XXX")x^2-9=0`

Factor the left side
`color(white)("XXX")(x+3)(x-3)=0`

which implies
`color(white)("XXX")x=-3color(white)("xxx") or color(white)("xxx")x=+3`

Checking for extraneous solutions:
#{: ("with "x=-3,,color(white)("xxx"),"with "x=+3,), (sqrt(5 * x^2-9)=+6,2x=-6,,sqrt(5^2-9)=+6,2x=6), ("extraneous",,,"valid",) :}#

The only valid solution is `x=+3`